Write the slope-intercept equation for the line that passes through (-3,-15)and is perpendicular to -6x+8y=3
First, let's solve the equation for y to determine the slope of the line.
-6x + 8y = 3
8y = 6x + 3
y = (6/8)x + 3/8 = (3/4)x + 3/8
Slope = 3/4
The slope of a line perpendicular to this one must be its negative reciprocal, or -4/3.
Now, use the point-slope form to find the other line:
y - y1 = m(x - x1)
y - (-15) = -4/3 [x - (-3)]
y + 15 = -4/3 (x + 3)
y + 15 = (-4/3)x - 4
y = (-4/3)x - 4 - 15
y = (-4/3)x - 19
And there you have it! By showing you the steps, I hope this will help with other problems of this type.
To write the slope-intercept equation for a line, we need to find the slope (m) and the y-intercept (b).
First, let's find the slope of the given line. The equation -6x + 8y = 3 is in standard form, which is not convenient for finding the slope. We can rewrite this equation in slope-intercept form (y = mx + b) by isolating y.
Starting with -6x + 8y = 3:
1. Add 6x to both sides: 8y = 6x + 3
2. Divide the entire equation by 8: y = (6/8)x + (3/8)
3. Simplify: y = (3/4)x + 3/8
Now we have the slope-intercept form of the given line, which is y = (3/4)x + 3/8. The slope of this line is 3/4.
Since the line we want to find is perpendicular to the given line, its slope will be the negative reciprocal of 3/4. The negative reciprocal of a fraction (p/q) is -q/p. Therefore, the slope of the line we're trying to find is -4/3.
Now, we have the slope (-4/3) and one point (-3, -15). We can apply the point-slope formula to find the equation.
The point-slope formula is y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Using (-3, -15) and m = -4/3, the equation becomes:
y - (-15) = -4/3(x - (-3))
Simplifying:
y + 15 = -4/3(x + 3)
Expanding:
y + 15 = -4/3x - 4/3(3)
y + 15 = -4/3x - 4/3(3/1)
y + 15 = -4/3x - 4/1
y + 15 = -4/3x - 12/3
y + 15 = -4/3x - 4
Rearranging the equation in slope-intercept form (y = mx + b):
y = -4/3x - 4 - 15
y = -4/3x - 19
Therefore, the slope-intercept equation for the line that passes through (-3, -15) and is perpendicular to -6x + 8y = 3 is y = -4/3x - 19.